Plain-language statement
"Ordinary" subadditivity of von Neumann entropy
Exact Lean statement
theorem Sᵥₙ_subadditivity (ρ : MState (d₁ × d₂)) :
Sᵥₙ ρ ≤ Sᵥₙ ρ.traceRight + Sᵥₙ ρ.traceLeftFormal artifact
Lean source
theorem Sᵥₙ_subadditivity (ρ : MState (d₁ × d₂)) : Sᵥₙ ρ ≤ Sᵥₙ ρ.traceRight + Sᵥₙ ρ.traceLeft := by have := Sᵥₙ_strong_subadditivity (ρ.relabel (d₂ := d₁ × Unit × d₂) ⟨fun x ↦ (x.1, x.2.2), fun x ↦ (x.1, ⟨(), x.2⟩), fun x ↦ by simp, fun x ↦ by simp⟩) simp [Sᵥₙ_relabel] at this convert this using 1 congr 1 · convert Sᵥₙ_relabel _ (Equiv.prodPUnit _).symm exact rfl · convert Sᵥₙ_relabel _ (Equiv.punitProd _).symm exact rfl- Project
- quantumInfo
- License
- MIT
- Commit
- 56e83a9288a3
- Source
- QuantumInfo/Finite/Entropy/SSA.lean:1191-1201
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Related declarations
Conj Transpose isometry mul isometry le one
conjTranspose_isometry_mul_isometry_le_one
Project documentation
The operator norm of the conjugate transpose is equal to the operator norm. -/ theorem Matrix.opNorm_conjTranspose_eq_opNorm {m n : Type*} [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] (A : Matrix m n 𝕜) : Matrix.opNorm Aᴴ = Matrix.opNorm A := by unfold Matrix.opNorm rw [← ContinuousLinearMap.adjoint.norm_map (toEuclideanLin A).toContinuousLine...
Source project: quantumInfo
Person-level attribution pending.
Convex roof of pure
convex_roof_of_pure
Plain-language statement
The convex roof extension of g : KetUpToPhase d → ℝ≥0 applied to a pure state ψ is g (KetUpToPhase.mk ψ).
Source project: quantumInfo
Person-level attribution pending.
Id achieves Rate log dim
CPTPMap.id_achievesRate_log_dim
Plain-language statement
The identity channel on D dimensional space achieves a rate of log2(D).
Source project: quantumInfo
Person-level attribution pending.